Limits of
it-soft sets and their applications for rough
sets
IXiaoliang Xie
(Mobile E-business Collaborative Innovation Center of Hunan Province, Hunan
University of Commerce, Changsha, Hunan 410205, P.R.China.)
Jiali He∗
(Key Laboratory of Complex System Optimization and Big Data Processing in Department of Guangxi Education, Yulin Normal University, Yulin, Guangxi 537000,
P.R.China)
Abstract
Soft set theory is a mathematical tool for dealing with uncertainty. This paper investigates limits of interval type of soft sets (for short, it-soft sets). The concept of it-soft sets is first introduced. Then, limits of it-soft sets are proposed and their properties are obtained. Next, point-wise continuity of it-soft sets and continuous it-soft sets are discussed. Finally, an application for rough sets is given.
Key words: Soft set; it-soft set; Limit; Continuity; Rough set.
1. Introduction
To solve complicated problems in economics, engineering, environmental science and social science, methods in classical mathematics are not always successful because of various types of uncertainties present in these prob-lems. There are several theories: probability theory, fuzzy set theory [22],
IThis work is supported by National Natural Science Foundation of China (11461005), Key Laboratory of Hunan Province for New Retail Virtual Reality Technology and Key Laboratory of Hunan Province for Mobile Business Intelligence.
∗Corresponding author: Jiali He
rough set theory [18] and the interval mathematics which we can consider as mathematical tools for dealing with uncertainties. But all these theories have their own difficulties. For example, probability theory can deal only with stochastically stable phenomena (see [17]). To overcome these kinds of difficulties, Molodtsov [17] proposed a completely new approach, which is called soft set theory, for modeling uncertainty.
Presently, works on soft sets theory are progressing rapidly. Maji et al. [14, 15] further studied soft sets theory and used this theory to solve some decision making problems. Aktas et al. [1] defined soft groups. Jiang et al. [7] extended soft sets with description logics. Feng et al. [4] investigated the relationship among soft sets, rough sets and fuzzy sets. Ge et al. [8] discussed the relationship between soft sets and topological spaces. Li et al. [12] obtained the relationship among soft sets, soft rough sets and topologies. Li et al. [13] studied parameter reductions of soft coverings.
Rough set theory, proposed by Pawlak [18], is an important tool for deal-ing with fuzzyness and uncertainty of knowledge. After thirty years de-velopment, this theory has been successfully applied to machine learning, intelligent systems, inductive reasoning, pattern recognition, mereology, im-age processing, signal analysis, knowledge discovery, decision analysis, expert systems and many other fields [18, 19, 20, 21]. The basic structure of rough set theory is an approximation space. Based on it, lower and upper approx-imations can be induced. Through these rough approxapprox-imations, knowledge hidden in information systems may be revealed and expressed in the form of decision rules [19, 20, 21]. Pawlak’s rough set model is based on the com-pleteness of available information, and ignores the incomcom-pleteness of available information and the possible existence of statistical information. This model for extracting rules in uncoordinate decision information systems often seems incapable. These have motivated many researchers to investigate probabilis-tic generalization of rough set theory and provide new rough set model for the study of uncertain information system.
Probabilistic rough set model is probabilistic generalization of rough set theory. In probabilistic rough set model, probabilistic rough approximations are dependent on parameters. Researching the infinite change trend or the limit state of these approximations accordance with parameters is helpful for the study of probabilistic rough sets.
level sets of a fuzzy set are both it-soft sets (i.e., interval type of soft sets or soft sets whose parameter sets are the intervals in R), we may attempt to study the infinite change trend or the limit state of it-soft sets. It is worth mentioning that there is no systematic research and summary for limits of it-soft sets although the limit though ofit-soft sets has formed in [24, 25].
In general, most of uncertain mathematical theories can only deal with uncertainty problems of discreteness. If limit theory of it-soft sets is es-tablished, then these theories may be used to solve uncertainty problems of continuity The purpose of this paper is to establish preliminarily limit theory of interval type soft set so that some uncertain mathematical theories such as rough set theory may be used to solve uncertainty problems of continuity. The remaining part of this paper is organized as follows. In Section 2, we recall some basic concepts about limits of set sequences and rough sets. In Section 3, we introduce it-soft sets and related notions. In Sections 4, we propose the concept of limits of it-soft sets and obtain their properties. In Sections 5, we discuss the continuity of it-soft sets including point-wise continuity of it-soft sets and continuous it-soft sets. In Sections 6, we give an application for rough sets. Sections 7 summarizes this paper.
2. Preliminaries
In this section, we recall some basic concepts about limits ofs-sequences, rough sets and it-soft sets.
Throughout this paper,U denotes the universe which may be an infinite set, 2U denotes the family of all subsets ofU, E denotes a set of all possible
parameters, R denotes the set of all real numbers, N denotes the set of all natural numbers and I denotes the interval inR.
2.1. Limits of set sequences
Definition 2.1 ([3, 9]). Let U be the universe. If for each n∈N, En∈2U,
then {En} is called a set sequence in U. Define
lim
n→∞En={x∈U :{n ∈N :x∈En} is inf inite},
lim
n→∞
En ={x∈U :{n∈N :x /∈En} is f inite}.
If lim
n→∞
En = lim
n→∞En = E, then {En : n ∈ N} is called to has the limit E,
which is denoted by lim
n→∞En, i.e., nlim→∞En = E; If nlim→∞En ̸= limn→∞En, then
Obviously, lim
n→∞
En⊆ lim n→∞En.
Proposition 2.2 ([3, 9]). Let {En:n ∈N} be a set sequence in U.
(1) lim
n→∞En =
∞
∩
n=1 ∞
∪
k=n
Ek.
(2) lim
n→∞
En =
∞
∪
n=1 ∞
∩
k=n
Ek.
Proposition 2.3 ([3, 9]). Let {En:n ∈N} be a set sequence in U.
(1) If {En} ↑, then lim n→∞En =
∞
∪
n=1
En.
(2) If {En} ↓, then lim n→∞En =
∞
∩
n=1
En.
2.2. Rough sets
LetR be an equivalence relation on the universeU. Then the pair (U, R) is called a Pawlak approximation space. Based on (U, R), one can define the following two rough approximations:
R(X) = {x∈U : [x]R⊆X}, R(X) = {x∈U : [x]R∩X ̸=∅}.
Then R(X) and R(X) are called the Pawlak lower approximation and the Pawlak upper approximation of X, respectively.
The boundary region ofX, defined by the difference between these rough approximations, that is BndR(X) =R(X)−R(X).
A set is rough if its boundary region is not empty; otherwise, it is crisp. Thus, X is rough if R(X)̸=R(X).
Definition 2.4 ([24, 25]). Let U be a finite universe. Then a function P : 2U → [0,1] is called a probability measure over U, if P(U) = 1 and P(A∪
B) =P(A) +P(B) whenever A∩B =∅.
If P is a probability measure over U, A, B ∈ 2U and P(B) > 0, then P(A|B) = PP(A(B∩B)) is called the conditional probability of the event A when the event B occurs.
upper approximation of X, are respectively denoted byP Iα(X)and P Iβ(X),
are defined as follows:
P Iα(X) ={x∈U :P(X|[x])≥α}, P Iβ(X) = {x∈U :P(X|[x])> β),
where 0≤β < α≤1.
Theorem 2.6 ([24, 25]). Let (U, R, P) be a probabilistic approximate space. Then the following properties hold.
(1) P Iα(∅) =P Iα(∅) =∅, P Iα(U) =P Iα(U) = U.
(2) P Iα(X)⊆P Iα(X).
(3) P Iα(U −X) =U −P I1−α(X), P Iα(U −X) =U −P I1−α(X).
(4) If X ⊆Y, then P Iα(X)⊆P Iα(Y), P Iα(X)⊆P Iα(Y).
(5) If 0< α1 < α2 ≤1, 0≤β1 < β2 <1 then
P Iα
2(X)⊆P Iα1(X), P Iβ2(X)⊆P Iβ2(X).
Theorem 2.7 ([24, 25]). Let (U, R, P) be a probabilistic approximate space. Then for 0< γ <1, X ∈2U,
(1) lim
α↑γP Iα(X) =
∩
α∈(0,γ)
P Iα(X) =P Iγ(X),
lim
α↓γP Iα(X) =
∪
α∈(γ,1]
P Iα(X) =P Iγ(X);
(2) lim
α↑γP Iα(X) =
∩
α∈[0,γ)
P Iα(X) =P Iγ(X),
lim
α↓γP Iα(X) =
∪
α∈(γ,1)
P Iα(X) = P Iγ(X).
Although the limit though ofit-soft sets has formed in Theorem 2.6, there is no systematic research and summary for limits of it-soft sets. Thus, limit theory of interval type soft set deserves deeply study so that rough set theory can be used to solve uncertainty problems of continuity.
3. Soft sets
Definition 3.1 ([17]). Let A⊆E. A pair (f, A) is called a soft set over U, if f is a mapping given by f :A→2U. We also denote (f, A) by f
A.
In other words, a soft setfAoverU is a parametrized family of subsets of
Definition 3.2 ([14]). Let fA and gB be two soft sets over U.
(1) fA is called a soft subset of gB, if A ⊆ B and f(e) = g(e) for each
e∈A. We denote it by fA ⊂e gB.
(2) fA is called a soft super set of gB, if gB ⊂e fA. We denote it by
fA ⊃e gB.
Definition 3.3 ([14]). Let fA and gB be two soft sets over U.
fA and gB are called soft equal, ifA⊆B and f(e) = g(e)for each e∈A.
We denote it by fA=gB.
Obviously, fA=gB if and only if fA ⊂e gB and fA ⊃e gB.
Definition 3.4 ([14]). Let fA be a soft set over U.
(1) fA is called null, if f(e) = ∅ for each e∈A. We denote it by e∅.
(2) fA is called absolute, if f(e) = U for each e∈A. We denote it by Ue.
(3) fA is called constant, if there exists X ∈ 2U such that f(e) =X for
each e∈A. We denote it by Xe or XA.
Definition 3.5 ([14]). Let fA and gB be two soft sets over U.
(1) hC is called the intersection of fA and gB, if C =A∩B and h(e) =
f(e)∩g(e) for each e∈C. We denote it by fA ∩e gB =hC.
(2) hC is called the union of fA and gB, if C =A∪B and
h(e) =
f(e), if e∈A−B, g(e), if e∈B−A, f(e)∪g(e), if e∈A∩B.
We denote it by fA ∪e gB =hC.
(3) hC is called the bi-intersection of fA and gB, if C = A ×B and
h(a, b) =f(a)∩g(b) for each a∈A and b∈B. We denote it by fA
∧
gB=
hC.
(4) hC is called the bi-union of fA and gB, if C = A×B and h(a, b) =
f(a)∪g(b) for each a∈A and b∈B. We denote it by fA
∨
gB =hC.
Definition 3.6 ([16]). The relative complement of a soft set fA is denoted
by fc
A and is defined by fcA = (fc, A), where fc : A → 2U is a mapping
Definition 3.7 ([4]). Let fA be a soft set over U.
(1) fA is called full, if
∪
e∈A
f(e) =U.
(2) fA is called partition, if {f(e) :e∈A} forms a partition of U.
Definition 3.8 ([12]). Let fA be a soft set over U.
(1) fA is called topological, if {f(e) :e∈A} is a topology on U.
(2)fAis called keeping intersection, if for anya, b∈A, there existsc∈A
such that f(a)∩f(b) = f(c).
(2) fA is called keeping union, if for any a, b ∈A, there existsc∈Asuch
that f(a)∪f(b) = f(c).
(3) fA is called perfect, iff :A→2U is onto.
(4) fA is called having no kernel, if ∩{f(e) :e∈A}=∅.
Definition 3.9. Let fA be a soft set over U.
(1)fAis called strong keeping intersection, if for each B ⊆A, there exists
b ∈A such that ∩
a∈A
f(a) = f(b).
(2)fAis called strong keeping union, if for eachB ⊆A, there existsb∈A
such that ∪
a∈A
f(a) = f(b).
Obviously,fAis strong keeping intersection ⇒fAis keeping intersection,
fA is strong keeping union ⇒fA is keep union.
Proposition 3.10 ([12]). Let fA be a soft set over U. Then the following
properties hold.
(1) If fA is topological, then fA is full, keeping intersection and strong
keep union.
(2) fA is perfect if and only if {f(e) :e∈A} is a discrete topology over
U.
(3) If fA is perfect, then fA is topological.
(4) fA is having no kernel if and only if (fc, A) is full.
Example 3.11. LetU ={x1, x2, x3, x4, x5}, A= [0,1). DefinefAas follows:
f(e) =
{x1, x2, x5}, if α ∈[0,14), ∅, if α ∈[14,12),
{x1, x2}, if α ∈[12,34),
U, if α ∈[34,1).
Example 3.12. LetU ={x1, x2, x3, x4, x5}, A= [0,1). DefinefAas follows:
f(e) =
{x1, x2, x5}, if α ∈[0,14), {x1, x2}, if α ∈[14,12), {x3}, if α ∈[12,34), {x3, x4}, if α ∈[34,1).
Note that {x1, x2, x5} ∩ {x3} = ∅ ̸= f(α) (∀ α ∈ I). Then fA is not
keeping intersection.
Example 3.13. LetU ={x1, x2, x3, x4, x5}, A= [0,1). DefinefAas follows:
f(e) =
{x1}, if α ∈[0,14), {x1, x4}, if α ∈[14,12), {x1, x3, x4}, if α ∈[12,34),
U, if α ∈[34,1).
Then fA is full, keeping intersection and strong keeping union. But fA is not
topological.
Example 3.14. LetU ={x1, x2, x3, x4, x5}, A= [0,1). DefinefAas follows:
f(e) =
{x1, x2}, if α∈[0,14), {x5}, if α∈[14,12), {x3}, if α∈[12,34), {x4}, if α∈[34,1).
Then fA is partition. But fA is neither topological nor perfect.
Example 3.15. LetU ={x1, x2, x3, x4, x5}, A= [0,1). DefinefAas follows:
f(e) =
{x1, x2, x5}, if α ∈[0,14), ∅, if α ∈[14,12),
{x3}, if α ∈[12,34), {x3, x4}, if α ∈[34,1).
Then fA is full and strong keeping intersection. But
{x1, x2, x5} ∪ {x3}={x1, x2, x3, x5} ̸=f(α) (∀ α∈I).
Example 3.16. LetU ={x1, x2, x3, x4, x5}, A= [0,1). DefinefAas follows:
f(e) =
{x1}, if α∈[0,14), {x2}, if α∈[14,12), {x1, x2}, if α∈[12,34),
U, if α∈[34,1).
Then fA is full and strong keeping union. But
{x1} ∩ {x2}=∅ ̸=f(α) (∀ α∈I).
Thus fA is not keeping intersection.
From Examples 3.11, 3.12, 3.13, 3.14, 3.15 and 3.16, we have the following relationships:
f is full and keeping intersection f is full and strong keeping union f is topological
f is full, keeping intersection and strong keeping union
fA is perfect
fA is partition
fA is topological
4. Limit theory of it-soft sets
4.1. The concept of it-soft sets
Definition 4.1. Let fA be a soft set over U. If there exists the interval I in
R such that A =I. Then fA is called an it-soft set over U. Denote it with
It is worth mentioning that theit-soft sets are different from interval soft sets in [23].
Definition 4.2. Let fI be an it-soft set over U.
(1) If for any e1, e2 ∈ I, e1 < e2 implies f(e1) ⊂ f(e2)(resp., f(e1) ⊃
f(e2)), then fI is called strictly increasing (resp., strictly decreasing) on I.
(2) If for any e1, e2 ∈ I, e1 < e2 implies f(e1) ⊆ f(e2)(resp., f(e1) ⊇
f(e2)), then fI is called increasing (resp., decreasing) on I.
Definition 4.3. Let fI be an it-soft set over U.
(1) If for any e ∈ I, f(e) ⊆ f(e0) (e0 ∈ I), then f(e0) is called the
maximum value of fI.
(2) If for any e ∈ I, f(e) ⊇ f(e0) (e0 ∈ I), then f(e0) is called the
minimum value of fI.
4.2. Limits of it-soft sets
Lete0 ∈R,δ > 0. Denote
U(e0, δ) ={e:|e−e0|< δ}, U0(e0, δ) ={e: 0<|e−e0|< δ}.
Then U(e0, δ) is called δ neighborhood of e0, U0(e0, δ) is called δ
neighbor-hood ofe0 having no the heart,e0 is the center of the neighborhood, δ is the
radius of the neighborhood. U+(e
0, δ) = [e0, e0+δ) is called the δ right neighborhood ofe0,
U−(e0, δ) = (e0−δ, e0] is called the δ left neighborhood of e0.
Obviously, U(e0, δ) = (e0−δ, e0+δ) =U+(e0, δ)∪U−(e0, δ).
LetfI be an it-soft set over U. For e0 ∈I, x∈U, denote
[x]fI ={e∈I− {e0}:x∈f(e)},
(x)fI ={e∈I− {e0}:x /∈f(e)}.
Remark 4.4. (1) [x]fI ∪(x)fI =I− {e0}, [x]fI∩e(x)fI =∅.
(2) [x]fI ∩[x]gI = [x]fI∩egI, [x]fI ∪ [x]gI = [x]fI∪egI.
(3) (x)fI ∩(x)gI = (x)fIe∪gI, (x)fI ∪(x)gI = (x)fI∩egI.
(4) [x]fc
I = (x)fI, (x)f c
I = [x]fI.
Definition 4.5. Let fI be an it-soft set over U. For e0 ∈I, define
(1) lim
e→e+0
f(e) = {x∈U :∀ δ >0, [x]fI ∩U
+(e
0, δ) is inf inite}, which is
(2) lim
e→e+0
f(e) = {x ∈ U : ∃ δ > 0, (x)fI ∩U
+(e
0, δ) is f inite}, which is
called the under-right limit of fI as e → e0(or the under limit of fI as e →
e+0).
(3) lim
e→e−0
f(e) ={x∈U :∀ δ >0, [x]fI ∩U
−(e
0, δ) is inf inite}, which is
called the over-left limit of fI as e→e0(or the over limit offI as e→e−0).
(4) lim
e→e−0
f(e) = {x ∈ U : ∃ δ > 0, (x)fI ∩U−(e0, δ) is f inite}, which is
called the under-left limit of fI as e → e0(or the under limit of fI as e →
e−0).
Theorem 4.6. Let fI be an it-soft set over U. Then for e0 ∈I,
(1) lim
e→e+0
f(e) = {x∈U :∀ δ >0, [x]fI ∩U
+(e
0, δ)̸=∅}
={x∈U :∀ n∈N, [x]fI ∩U
+(e
0,n1)̸=∅}.
(2) lim
e→e+0
f(e) = {x∈U :∃ δ >0, (x)fI ∩U
+(e
0, δ) = ∅}
={x∈U :∃ n∈N, (x)fI ∩U
+(e
0,n1) =∅}.
(3) lim
e→e−0
f(e) ={x∈U :∀ δ >0, [x]fI ∩U
−(e
0, δ)̸=∅}
={x∈U :∀ n∈N, [x]fI ∩U
−(e
0,n1)̸=∅}.
(4) lim
e→e−0
f(e) ={x∈U :∃ δ >0, (x)fI ∩U
+(e
0, δ) = ∅}
={x∈U :∃ n∈N, (x)fI ∩U−(e0,
1
n) =∅}.
Proof. (1) Put
S= lim
e→e+0
f(e), T ={x∈U :∀ δ >0, [x]fI ∩U
+(e
0, δ)̸=∅},
L={x∈U :∀ n ∈N, [x]fI ∩U
+(e
0,n1)̸=∅}.
Obviously, S ⊆ T ⊆ L. We only need to prove L ⊆ S. Suppose L * S. Then L− S ̸= ∅. Pick x ∈ L− S. We have x ̸∈ S. So ∃ δ0 > 0,
[x]fI ∩U
+(e
0, δ0) is finite. Denote
[x]fI ∩U
+
(e0, δ0) ={e1, e2, . . . , en}.
Pute∗ = min{e1, e2, . . . , en}, 0< n1
0 < e
∗−e
0. Then
0< 1 n0
< δ0, [x]fI ∩U
+(e 0,
1 n0
) =∅.
(2) Put
P = lim
e→e+0
f(e), Q={x∈U :∃ δ >0, (x)fI ∩U
+(e
0, δ) =∅},
K ={x∈U :∃ n∈N, (x)fI ∩U
+(e
0,n1) = ∅}.
Obviously,K ⊆Q⊆P. We only need to proveP ⊆K. SupposeP *K. Then P −K ̸=∅. Pick x∈P −K. Then x /∈K.
Claim ∀δ, (x)fI ∩U
+(e
0, δ) is infinite.
In fact, suppose that∃ δ >0, (x)fI ∩U
+(e
0, δ) is finite. Put
(x)fI∩U
+(e
0, δ) ={e1, e2, . . . , en}, e∗ = min{e1, e2, . . . , en}, 0<
1 n0
< e∗−e0.
Then 0 < n1
0 < δ, (x)fI ∩U
+(e
0,n10) =∅. So x∈ K, But x̸∈ K. This is a
contradiction.
Since ∀ δ > 0, (x)fI ∩U
+(e
0, δ) is infinite, we have x ̸∈ P. But x ∈ P.
This is a contradiction. Thus P ⊆K. (3) The proof is similar to (1). (4) The proof is similar to (2).
Example 4.7. Consider Example 3.12, pick e0 = 14, we have
[x1]f = [x2]f = [0,
1 4)∪[
1 4,
1
2), [x3]f = [ 1
2,1), [x4]f = [ 3
4,1), [x5]f = [0, 1 4).
(x1)f = (x2)f = [
1
2,1), (x3)f = [0, 1 4)∪[
1 4,
1
2), (x4)f = [0, 1 4)∪[
1 4,
3
4), (x5)f = ( 1 4,1).
By Theorem 4.6,
lim
e→e+0
f(e) ={x∈U :∀ δ >0, [x]fI ∩U
+
(e0, δ)̸=∅}={x1, x2};
lim
e→e+0
f(e) ={x∈U :∃ δ >0, (x)fI ∩U
+(e
0, δ) =∅}={x1, x2};
lim
e→e−0
f(e) ={x∈U :∀ δ >0, [x]fI ∩U
−(e
0, δ)̸=∅}={x1, x2, x5};
lim
e→e−0
f(e) ={x∈U :∃ δ >0, (x)fI ∩U
−(e
Lemma 4.8. Let fI be an it-soft set over U. Then for e0 ∈I,
(1) lim
e→e+0
f(e) = ∩∞
n=1
∩
e∈(e0,e0+1n)∩I
∪
β∈(e0,e] f(β).
(2) lim
e→e+0
f(e) = ∪∞
n=1
∪
e∈(e0,e0+1n)∩I
∩
β∈(e0,e] f(β).
(3) lim
e→e−0
f(e) = ∩∞
n=1
∩
e∈(e0−n1,e0)∩I
∪
β∈[e,e0) f(β).
(4) lim
e→e−0
f(e) = ∪∞
n=1
∪
e∈(e0−n1,e0)∩I
∩
β∈[e,e0) f(β).
Proof. (1) Denote
S = lim
e→e+0
f(e), T =
∞
∩
n=1
∩
e∈(e0,e0+n1)∩I
∪
β∈(e0,e] f(β).
To prove S =T, it suffices to show that
x∈S ⇔ ∀ n∈N, ∀ e∈(e0, e0+
1
n)∩I, ∃ β ∈(e0, e], x∈f(β).
“ ⇒ ”. Let x ∈ S, ∀ n ∈ N, ∀ e ∈ (e0, e0 + n1)∩I. Put δ = e−e0.
Then 0< δ < n1.
Since x ∈ S, by Theorem 4.6(1), we have [x]fI ∩ U
+(e
0, δ) ̸= ∅. Pick
β ∈ [x]fI ∩U
+(e
0, δ). Thenβ ∈ [x]fI, β ∈U
+(e 0, δ).
This impliesx∈f(β), e0 < β < e0+δ =e. Thus β ∈(e0, e].
“⇐”. ∀ n∈N, pick e∈(e0, e0+n1)∩I.
By the condition, ∃β ∈(e0, e], x∈f(β).Thenβ ∈U+(e0,n1), β∈ [x]fI.
Thus ∀ n∈N, [x]fI ∩U
+(e
0,1n)̸=∅.
By Theorem 4.6(1), x∈S. (2) By (1) and Theorem 4.6(2),
x /∈ lim
e→e+0
f(e)
⇐⇒ ∀ n∈N, (x)fI ∩U
+(e
0,1n)̸=∅
⇐⇒ ∀ n∈N,{e∈I−e0 :x∈U −f(e)} ∩U+(e0,n1)̸=∅ ⇐⇒x∈ ∩∞
n=1
∩
e∈(e0,e0+1n)∩I
∪
β∈(e0,e]
(U −f(β))
⇐⇒x∈U − ∪∞
n=1
∪
e∈(e0,e0+n1)∩I
∩
⇐⇒x /∈ ∪∞
n=1
∪
e∈(e0,e0+1n)∩I
∩
β∈(e0,e] f(β).
Hence lim
e→e+0
f(e) = ∪∞
n=1
∪
e∈(e0,e0+n1)∩I
∩
β∈(e0,e] f(β).
(3) The proof is similar to (1). (4) The proof is similar to (2).
Lemma 4.9. Let fI be an it-soft set over U. Then for e0 ∈I,
(1) ∩∞
n=1
∩
e∈(e0,e0+n1)∩I
∪
β∈(e0,e]
f(β) = ∩
e∈(e0,e0+1)∩I
∪
β∈(e0,e] f(β).
(2) ∪∞
n=1
∪
e∈(e0,e0+n1)∩I
∩
β∈(e0,e]
f(β) = ∪
e∈(e0,e0+1)∩I
∩
β∈(e0,e] f(β).
(3) ∩∞
n=1
∩
e∈(e0−n1,e0)∩I
∪
β∈[e,e0)
f(β) = ∩
e∈(e0−1,e0)∩I
∪
β∈[e,e0) f(β).
(4) ∪∞
n=1
∪
e∈(e0−n1,e0)∩I
∩
β∈[e,e0)
f(β) = ∪
e∈(e0−1,e0)∩I
∩
β∈[e,e0) f(β).
Proof. (1) Put En=
∩
e∈(e0,e0+1n)∩I
∪
β∈(e0,e]
f(β). Then {En} ↑. So
∞
∩
n=1
En=E1.
Thus
∞
∩
n=1
∩
e∈(e0,e0+n1)∩I
∪
β∈(e0,e]
f(β) = ∩
e∈(e0,e0+1)∩I
∪
β∈(e0,e] f(β).
(2) PutFn=
∪
e∈(e0,e0+n1)∩I
∩
β∈(e0,e]
f(β). Then{Fn} ↓. So
∞
∪
n=1
Fn=F1.
Thus
∞
∪
n=1
∪
e∈(e0,e0+n1)∩I
∩
β∈(e0,e]
f(β) = ∪
e∈(e0,e0+1)∩I
∩
β∈(e0,e] f(β).
(3) The proof is similar to (1). (4) The proof is similar to (2).
Theorem 4.10. Let fI be an it-soft set over U. Then for e0 ∈I,
(1) lim
e→e+0
f(e) = ∩
e∈(e0,e0+1)∩I
∪
β∈(e0,e]
f(β); If fI increasing, then
lim
e→e+0
f(e) = ∩
(2) lim
e→e+0
f(e) = ∪
e∈(e0,e0+1)∩I
∩
β∈(e0,e]
f(β); If fI decreasing, then
lim
e→e+0
f(e) = ∪
e∈(e0,e0+1)∩I f(e).
(3) lim
e→e−0
f(e) = ∩
e∈(e0−1,e0)∩I
∪
β∈[e,e0)
f(β); If fI decreasing, then
lim
e→e−0
f(e) = ∩
e∈(e0−1,e0)∩I f(e).
(4) lim
e→e−0
f(e) = ∪
e∈(e0−1,e0)∩I
∩
β∈[e,e0)
f(β); If fI increasing, then
lim
e→e−0
f(e) = ∪
e∈(e0−1,e0)∩I f(e).
Proof. This holds by Lemmas 4.8 and 4.9.
Definition 4.11. Let fI be an it-soft set over U. Then for e0 ∈I,
(1) If lim
e→e+0
f(e) = lim
e→e+0
f(e) =S, then fI is called to has the limit S as
e → e+0 (or has the right-limit S as e → e0), which is denoted by lim
e→e+0
f(e),
i.e., lim
e→e+0
f(e) = S;
If lim
e→e+0
f(e) ̸= lim
e→e+0
f(e), then fI is called to has no the limit as e → e+0
(or has no the right-limit as e →e0).
(2) If lim
e→e−0
f(e) = lim
e→e−0
f(e) = S, then fI is called to has the limit S as
e → e−0 (or has the left-limit S as e → e0), which is denoted by lim
e→e−0
f(e),
i.e., lim
e→e−0
f(e) =S;
If lim
e→e+0
f(e) ̸= lim
e→e+0
f(e), then fI is called to has no the limit as e → e+0
(or has no the left-limit as e→e0).
(3) If lim
e→e−0
f(e) = lim
e→e+0
f(e) = S, then fI is called to has the limit S as
e→e0, which is denoted by lim
e→e0
f(e), i.e., lim
e→e0
f(e) = S;
If lim
e→e−0
f(e)̸= lim
e→e+0
Definition 4.12. Let fI be an it-soft set over U. Then for e0 ∈I,
(1) If lim
e→e−0
f(e) = lim
e→e+0
f(e) = S, then fI is called to has the over-limit
S as e→e0, which is denoted by lim
e→e0
f(e), i.e., lim
e→e0
f(e) =S;
If lim
e→e−0
f(e) ̸= lim
e→e+0
f(e), then fI is called to has no the over-limit as
e→e+0. (2) If lim
e→e−0
f(e) = lim
e→e+0
f(e) =S, then fI is called to has the under-limit
S as e→e0, which is denoted by lim
e→e0
f(e), i.e., lim
e→e0
f(e) =S;
If lim
e→e−0
f(e) ̸= lim
e→e+0
f(e), then fI is called to has no the under-limit as
e→e0.
(3) If lim
e→e0
f(e) = lim
e→e0
f(e) = S, then fI is called to has the limit as
e→e0, which is denoted by lim
e→e0
f(e), i.e., lim
e→e0
f(e) = S;
If lim
e→e0
f(e)̸= lim
e→e0
f(e), then fI is called to has no the limit as e→e0.
Remark 4.13. The limit in Definition 4.11(3) and the limit in Definition 4.12(3) is consistent.
Example 4.14. Let XI be a constantit-soft set overU whereX ∈2U. Then
for e0 ∈I, lim
e→e0
X(e) =X.
Obviously, [x]XI =
{
I− {e0}, x∈X
∅, x /∈X , (x)XI =
{
I− {e0}, x̸∈X
∅, x∈X .
By Theorem 4.6,
lim
e→e+0
X(e) = {x∈U :∀ δ >0, [x]Ae∩U+(e0, δ)̸=∅},
lim
e→e+0
X(e) = {x∈U :∃ δ >0, (x)Ae∩U+(e0, δ) = ∅}.
Then lim
e→e+0
X(e) =X, lim
e→e+0
X(e) =X.
Similarly, lim
e→e−0
X(e) = X, lim
e→e−0
X(e) = X.
Thus lim
e→e0
X(e) =X.
Definition 4.15. Let (f,(−∞,+∞)) be an it-soft set over U. Define
(1) lim
e→+∞f(e) = lime→0+f( 1
e), e→−∞lim f(e) = lime→0−f(
1 e),
lim
e→∞f(e) = lime→0f(
1 e).
(2) lim
e→+∞
f(e) = lim
e→0+ f(1
e), e→−∞lim
f(e) = lim
e→0−
f(1 e),
lim
e→∞
f(e) = lim
e→0
f(1 e).
(3) lim
e→+∞f(e) = lime→0+f( 1
e), e→−∞lim f(e) = lime→0−f(
1 e),
lim
e→∞f(e) = lime→0f(
1 e).
4.3. Properties of limits of it-soft sets
Proposition 4.16. For the over-right limit, the following properties hold: (1) If f(e)⊆g(e)(∀ e∈(e0, e0+δ0)), then lim
e→e+0
f(e)⊆ lim
e→e+0
g(e).
(2) lim
e→e+0
(f(e)∪g(e)) = lim
e→e+0
f(e)∪ lim
e→e+0
g(e).
(3) lim
e→e+0
(U −f(e)) =U − lim
e→e+0
f(e).
(4) If lim
e→e+0
f(e) =△ ⊂B, then ∃ δ >0,∀ e∈(e0, e0+δ), f(e)⊂B.
(5) 1) lim
e→e+0
(f(e)×g(e))⊆ lim
e→e+0
f(e)× lim
e→e+0
g(e);
2) lim
e→e+0
f(e)× lim
e→e+0
g(e) = ∩
e∈(e0,e0+1)∩I
∪
β,γ∈(e0,e]
(f(β)×g(γ)).
Proof. (1) Denote
[x]fI ={e∈I− {e0}:x∈f(e)}, [x]gI ={e∈I − {e0}:x∈g(e)}.
∀ x ∈ lim
e→e+0
f(e), by Theorem 4.6(1), ∀ δ > 0, [x]fI ∩U
+(e
0, δ) ̸= ∅.
Pick eδ∈[x]fI ∩U
+(e
0, δ). Thenx∈f(eδ),eδ ∈U+(e0, δ).
1) If δ ≤ δ0, then eδ ∈ U+(e0, δ0). By the condition, f(eδ) ⊆ g(eδ).
Thenx∈g(eδ). This implieseδ ∈(x)fI∩U
+(e
0, δ). So (X)fI∩U
+(e
2) Ifδ > δ0, then U+(e0, δ0)⊆U+(e0, δ). So (x)fI∩U
+(e
0, δ0)⊆(X)fI ∩ U+(e0, δ). Since eδ0 ∈(X)fI ∩U
+(e
0, δ0), we have (x)fI ∩U
+(e
0, δ)̸=∅.
By 1) and 2), ∀ δ > 0, (x)fI ∩ U
+(e
0, δ) ̸= ∅. By Theorem 4.6(1),
x∈ lim
e→e+0
g(e).
Thus
lim
e→e+0
f(e)⊆ lim
e→e+0
g(e).
(2) “⊇”. This holds by (1). “⊆”. Suppose lim
e→e+0
(f(e)∪g(e))̸⊆ lim
e→e+0
f(e)∪ lim
e→e+0
g(e). Then
lim
e→e+0
(f(e)∪g(e))− lim
e→e+0
f(e)∪ lim
e→e+0
g(e)̸=∅.
Pick x∈ lim
e→e+0
(f(e)∪g(e))− lim
e→e+0
f(e)∪ lim
e→e+0
g(e). We have
x∈ lim
e→e+0
(f(e)∪g(e)), x /∈ lim
e→e+0
f(e) and x /∈ lim
e→e+0
g(e).
By Theorem 4.6,∃δ1, δ2 >0, [x]f∩U+(e0, δ1) =∅, [x]g∩U+(e0, δ2) =∅.
Pickδ3=min{δ1, δ2}. Then [x]f∩U+(e0, δ3) =∅and [x]g∩U+(e0, δ3) =∅.
It follows
([x]f ∪[x]gI)∩U
+(e
0, δ3) = ([x]f ∩U+(e0, δ3))∪([x]g∩U+(e0, δ3)) = ∅.
By Remark 4.4, [x]f∪g∩U+(e0, δ3) = ∅.
Thus x /∈ lim
e→e+0
(f ∪g)(e) = lim
e→e+0
(f(e)∪g(e)). This is a contradiction.
(3)∀x∈ lim
e→e+0
(U−f(e)). Thenx∈ lim
e→e+0
fc(e). By Theorem 4.6,∀δ >0,
[x]fc ∩U+(e0, δ)̸=∅. By Remark 4.4, (x)f ∩U+(e0, δ)̸=∅. Thus
x∈U − lim
e→e+0
f(e).
Conversely, the proof is similar.
(4) Suppose that ∀ δ >0, ∃ e∈(e0, e0+δ), f(e)*B or f(e) = B.
1) If f(e)*B, then f(e)−B ̸=∅. Pick x∈f(e)−B. We have
x∈f(e), x /∈B, e∈ [x]fI.
Sincee ∈(e0, e0+δ). Then [x]fI ∩(e0, e0+δ)̸=∅. So x∈ lim
e→e+0
Thusx∈B. This is a contradiction.
2) If f(e) = B, then △ −B =∅. So ∃ x∈B, x /∈ △.
Sincex∈f(e), we have x∈ [x]fI, [x]fI ∩(e0, e0+δ)̸=∅. So
x∈ lim
e→e+0
f(e) = △.
This is a contradiction. (5) 1) Put
Hf×g(e) =
∪
β∈(e0,e]
(f(β)×g(β)).
By Theorem 4.10(1),
lim
e→e+0
(f(e)×g(e)) = ∩
e∈(e0,e0+1)∩I
Hf×g(e).
∀(x, y)∈ lim
e→e+0
(f(e)×g(e)), we have (x, y)∈ ∩
e∈(e0,e0+1)∩I
Hf×g(e). Since
Hf×g(e) =
∪
β∈(e0,e]
(f(β)×g(β)),
we have∀e ∈(e0, e0+ 1)∩I, ∃βe∈(e0, e], (x, y)∈f(βe)×g(βe). It follows
x∈f(βe), y∈g(βe). Then x∈Hf(e) and y∈Hg(e). So
x∈ ∩
e∈(e0,e0+1)∩I
Hf(e) = lim e→e+0
f(e), y ∈ ∩
e∈(e0,e0+1)∩I
Hg(e) = lim e→e+0
g(e).
Thus (x, y)∈ lim
e→e+0
f(e)× lim
e→e+0
g(e).
Thus
lim
e→e+0
(f(e)×g(e))⊆ lim
e→e+0
f(e)× lim
e→e+0
g(e).
2)∀ (x, y)∈ lim
e→e+0
f(e)× lim
e→e+0
g(e), we have
x∈ lim
e→e+0
f(e) = ∩
e∈(e0,e0+1)∩I
∪
β∈(e0,e]
f(β), y ∈ lim
e→e+0
g(e) = ∩
e∈(e0,e0+1)∩I
∪
Then ∀ e ∈ (e0, e0 + 1) ∩ I, ∃ βe, γe ∈ (e0, e], x ∈ f(βe), y ∈ g(γe).
Then (x, y)∈f(βe)×g(γe). So
(x, y)∈ ∩
e∈(e0,e0+1)∩I
∪
β,γ∈(e0,e]
(f(β)×g(γ)).
Conversely, the proof is similar. Thus
lim
e→e+0
f(e)× lim
e→e+0
g(e) = ∩
e∈(e0,e0+1)∩I
∪
β,γ∈(e0,e]
(f(β)×g(γ)).
Proposition 4.17. For the under-right limit, the following properties hold. (1) If f(e)⊆g(e) (∀ e∈(e0, e0+δ0)), then lim
e→e+0
f(e)⊆ lim
e→e+0
g(e).
(2) lim
e→e+0
(f(e)∩g(e)) = lim
e→e+0
f(e)∩ lim
e→e+0
g(e).
(3) lim
e→e+0
(U −f(e)) =U − lim
e→e+0
f(e).
(4) If lim
e→e+0
f(e) =△ ⊃A, then ∃ δ >0, ∀ e ∈(e0, e0+δ), f(e)⊃A.
(5) lim
e→e+0
(f(e)×g(e)) = lim
e→e+0
f(e)× lim
e→e+0
g(e).
Proof. (1) The proof is similar to Proposition 4.16(1). (2) “⊆”. This holds by (1).
“⊇”. Suppose lim
e→e+0
f(e)∩lim
e→e+0
g(e)̸⊆ lim
e→e+0
(f(e)∩g(e)). Then lim
e→e+0
f(e)∩
lim
e→e+0
g(e)− lim
e→e+0
(f(e)∩g(e))̸=∅.Pickx∈ lim
e→e+0
f(e)∩ lim
e→e+0
g(e)− lim
e→e+0
(f(e)∩
g(e)). We have
x∈ lim
e→e+0
f(e),x∈ lim
e→e+0
g(e) and x /∈ lim
e→e+0
(f(e)∩g(e)).
By Theorem 4.6,
∃ δ1, δ2 >0, (x)f ∩U+(e0, δ1) = ∅, (x)g∩U+(e0, δ2) = ∅.
Pick δ3=min{δ1, δ2}. Then (x)f ∩U+(e0, δ3) = ∅, (x)g ∩U+(e0, δ3) = ∅.
It follows
((x)f ∪(x)gI)∩U
+(e
By Remark 4.4 , (x)f∩g∩U+(e0, δ3) = ∅.
Thus x∈ lim
e→e+0
(f ∩g)(e) = lim
e→e+0
(f(e)∩g(e)). This is a contradiction.
(3)∀x∈ lim
e→e+0
(U−f(e)). Thenx∈ lim
e→e+0
fc(e). By Theorem 4.6,∃δ >0,
(x)fc ∩U+(e0, δ) =∅. By Remark 4.4, [x]f ∩U+(e0, δ) = ∅.
Thusx∈U − lim
e→e+0
f(e).
Conversely, the proof is similar. (4) By Proposition 4.16(3),
lim
e→e+0
(U −f(e)) =U − lim
e→e+0
f(e).
Since lim
e→e+0
f(e) =△ ⊃A, we have lim
e→e+0
(U−f(e))⊂U −A.
By Proposition 4.16(4),∃ δ >0,∀e∈(e0, e0+δ),U −f(e)⊂U −A.
Thus
∃ δ >0,∀ e∈(e0, e0+δ), f(e)⊃A.
(5) ∀ (x, y)∈ lim
e→e+0
(f(e)×g(e)), by Theorem 4.10(2),
(x, y)∈ ∪
e∈(e0,e0+1)∩I
∩
β∈(e0,e]
(f(β)×g(β)).
Then ∃ e∈(e0, e0+ 1)∩I, ∀ β ∈(e0, e], (x, y)∈f(β)×g(β). It follows
x∈f(β), y∈g(β). Then
x∈ ∪
e∈(e0,e0+1)∩I
∩
β∈(e0,e]
f(β), y∈ ∪
e∈(e0,e0+1)∩I
∩
β∈(e0,e] g(β).
By Theorem 4.10(2),x∈ lim
e→e+0
f(e),y∈ lim
e→e+0
g(e). Thus (x, y)∈ lim
e→e+0
f(e)×
lim
e→e+0
g(e).
∀(x, y)∈ lim
e→e+0
f(e)× lim
e→e+0
g(e), By Theorem 4.10(2),
x∈ lim
e→e+0
f(e) = ∪
e∈(e0,e0+1)∩I
∩
β∈(e0,e]
f(β), y ∈ lim
e→e+0
g(e) = ∪
e∈(e0,e0+1)∩I
∩
Then ∃ e1, e2 ∈ (e0, e0 + 1)∩I, ∀ β ∈ (e0, e1], ∀ γ ∈ (e0, e2], x ∈ f(β),
y∈g(γ).
Pute∗ = min{e1, e2}. Thene∗ ∈(e0, e0+1)∩I, (e0, e∗]⊆(e0, e1]∩(e0, e2].
Then ∀ β ∈(e0, e∗], x∈f(β),y∈g(β). It follows (x, y)∈f(β)×g(β). So
(x, y)∈ ∪
e∈(e0,e0+1)∩I
∩
β,γ∈(e0,e]
(f(β)×g(β)).
By Theorem 4.10(2), (x, y)∈ lim
e→e+0
(f(e)×g(e)).
Thus
lim
e→e+0
(f(e)×g(e)) = lim
e→e+0
f(e)× lim
e→e+0
g(e).
Proposition 4.18. For the over-left limit, the following properties hold: (1) If f(e)⊆g(e) (∀ e∈(e0−δ0, e0)), then lim
e→e−0
f(e)⊆ lim
e→e−0
g(e).
(2) lim
e→e−0
(f(e)∪g(e)) = lim
e→e−0
f(e)∪ lim
e→e−0 g(e).
(3) lim
e→e−0
(U −f(e)) = U− lim
e→e−0
f(e).
(4) If lim
e→e−0
f(e) = △ ⊂B, then ∃ δ >0,∀ e∈(e0−δ, e0), f(e)⊂B.
(5) 1) lim
e→e−0
(f(e)×g(e))⊆ lim
e→e−0
f(e)× lim
e→e−0
g(e).
2) lim
e→e−0
f(e)× lim
e→e−0
g(e) = ∩
e∈(e0−1,e0)∩I
∪
β,γ∈[e,e0)
(f(β)×g(γ)).
Proof. The proof is similar to Proposition 4.16.
Proposition 4.19. For the under-left limit, the following properties hold: (1) If f(e)⊆g(e) (∀ e∈(e0−δ0, e0)), then lim
e→e−0
f(e)⊆ lim
e→e−0
g(e).
(2) lim
e→e−0
(f(e)∩g(e)) = lim
e→e−0
f(e)∩ lim
e→e−0
g(e).
(3) lim
e→e−0
(U −f(e)) = U− lim
e→e−0
f(e).
(4) If lim
e→e−0
f(e) = △ ⊃A, then ∃ δ >0,∀ e ∈(e0 −δ, e0), f(e)⊃A.
(5) lim
e→e−0
(f(e)×g(e)) = lim
e→e−0
f(e)× lim
e→e−0
Proof. The proof is similar to Proposition 4.17.
Corollary 4.20. Let fI be an it-soft set over U and A∈2U. For e0 ∈I,
(1) If f(e)⊆A or f(e)⊂A (∀ e∈(e0, e0+δ0)), then
lim
e→e+0
f(e)⊆A, lim
e→e+0
f(e)⊆A.
(2) If f(e)⊆A or f(e)⊂A (∀ e∈(e0−δ0, e0)), then
lim
e→e−0
f(e)⊆A, lim
e→e−0
f(e)⊆A.
Proof. This holds by Propositions 4.16, 4.17, 4.18, 4.19.
Corollary 4.21. Let fI be an it-soft set over U and A∈2U. For e0 ∈I,
(1) If f(e)⊇A or f(e)⊃A (∀ e∈(e0, e0+δ0)), then
lim
e→e+0
f(e)⊇A, lim
e→e+0
f(e)⊇A.
(2) If f(e)⊇A or f(e)⊃A (∀ e∈(e0−δ0, e0)), then
lim
e→e−0
f(e)⊇A, lim
e→e−0
f(e)⊇A.
Proof. This holds by Propositions 4.16, 4.17, 4.18, 4.19.
Theorem 4.22. For the over limit, the following properties hold: (1) If f(e)⊆g(e) (∀ e∈U0(e
0, δ0)), then lim
e→e0
f(e)⊆ lim
e→e0 g(e).
(2) lim
e→e0
(f(e)∪g(e)) = lim
e→e0
f(e)∪ lim
e→e0 g(e).
(3) lim
e→e0
(U −f(e)) =U − lim
e→e0 f(e).
(4) If lim
e→e0
f(e) =△ ⊂B, then ∃ δ > 0,∀ e∈U0(e
0, δ), f(e)⊂B.
(5) lim
e→e0
(f(e)×g(e))⊆ lim
e→e0
f(e)× lim
e→e0 g(e).
Proof. This holds by Propositions 4.16 and 4.18.
Theorem 4.23. For the under limit, the following properties hold: (1) If f(e)⊆g(e) (∀ e∈U0(e
0, δ0)), then lim
e→e0
f(e)⊆ lim
